2024 AMC 12B Problem 13

Attempt Problem 13 of the 2024 AMC 12B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2024 AMC 12B solutions, or check the answer key.

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13.

There are real numbers

x,

y,

h,

and

k

that satisfy the system of equations

x^2 + y^2 - 6x - 8y = h

x^2 + y^2 - 10x + 4y = k.

What is the minimum possible value of

h + k?

-54

-46

-34

-16

16

Add the two equations to get

h + k

as a single expression in

x

and

y

Complete the square:

h + k = 2(x-4)^2

+ 2(y-1)^2 - 34,

which is smallest when both squares vanish

Adding the equations,

\begin{aligned} h + k &= 2x^2 + 2y^2 - 16x - 4y \\ &= 2(x - 4)^2 \\ &\quad {}+ 2(y - 1)^2 - 34. \end{aligned}

Both squared terms are nonnegative, so the minimum occurs at

x = 4,

y = 1,

giving

h + k = -34.

Thus, the correct answer is C.

Problem 13 in other years